Local Fano-Mori contractions of high nef-value

Let $X$ be a variety with at most terminal $\mathbb Q$-factorial singularities of dimension $n$. We study local contractions $f:X\to Z$ supported by a $\mathbb Q$-Cartier divisor of the type $K_X+ \tau L$, where $L$ is an $f$-ample Cartier divisor and $\tau \geq 0$ is a rational number. Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in $\overline {NE(X)}_{K_X+\tau L = 0}$; these maps naturally arise in the context of the minimal model program. We prove that, if $\tau > (n-3) >0$, the general element $X' \in |L|$ is a variety with at most terminal singularities. Then we apply this to characterize, via an inductive argument, some birational contractions as above with $\tau > (n-3)\geq 0$.